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1: Software Index

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Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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2: 18.6 Symmetry, Special Values, and Limits to Monomials

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3: About the Project

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Refer to caption — Figure 1: The Editors and 9 of the 10 Associate Editors of the DLMF Project (photo taken at 3rd Editors Meeting, April, 2001). The front row, from left to right: Ronald F. …The back row, from left to right: William P. …

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4: 18.3 Definitions

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. In the second row

\mathcal{A}_{n}

denotes

\ifrac{2^{\alpha+\beta+1}\Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1% \right)}{((2n+\alpha+\beta+1)\Gamma\left(n+\alpha+\beta+1\right)n!)}

, with

\mathcal{A}_{0}=\ifrac{2^{\alpha+\beta+1}\Gamma\left(\alpha+1\right)\Gamma% \left(\beta+1\right)}{\Gamma\left(\alpha+\beta+2\right)}

. …

► ►►

Name	$p_{n}(x)$	$(a,b)$	$w(x)$	$h_{n}$	$k_{n}$	$\ifrac{\tilde{k}_{n}}{k_{n}}$	Constraints
…

► …

5: 24.14 Sums

… ►Let

\det[a_{r+s}]

denote a Hankel (or persymmetric) determinant, that is, an

(n+1)\times(n+1)

determinant with element

a_{r+s}

in row

r

and column

s

for

r,s=0,1,\dots,n

. … ►

24.14.11

\det[B_{r+s}]=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{6}\Bigg{/}\left(% \prod_{k=1}^{2n+1}k!\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

►

24.14.12

\det[E_{r+s}]=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{2}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

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6: 26.15 Permutations: Matrix Notation

… ►The set

\mathfrak{S}_{n}

(§26.13) can be identified with the set of

n\times n

matrices of 0’s and 1’s with exactly one 1 in each row and column. The permutation

\sigma

corresponds to the matrix in which there is a 1 at the intersection of row

j

with column

\sigma(j)

, and 0’s in all other positions. … ►The matrix represents the placement of

n

nonattacking rooks on an

n\times n

chessboard, that is, rooks that share neither a row nor a column with any other rook. … ►If

B=B_{1}\cup B_{2}

, where no element of

B_{1}

is in the same row or column as any element of

B_{2}

, then …

7: 34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent

\mathit{9j}

symbols. Even (cyclic) permutations of either columns or rows, as well as transpositions, leave the

\mathit{9j}

symbol unchanged. Odd permutations of columns or rows introduce a phase factor

(-1)^{R}

, where

R

is the sum of all arguments of the

\mathit{9j}

symbol. …

8: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►An

n

th-order determinant expanded by its

j

th row is given by …If two rows (or columns) of a determinant are interchanged, then the determinant changes sign. If two rows (columns) of a determinant are identical, then the determinant is zero. If all the elements of a row (column) of a determinant are multiplied by an arbitrary factor

\mu

, then the result is a determinant which is

\mu

times the original. If

\mu

times a row (column) of a determinant is added to another row (column), then the value of the determinant is unchanged. …

9: 26.12 Plane Partitions

… ►A plane partition,

\pi

, of a positive integer

n

, is a partition of

n

in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. … ►A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns. … ►A descending plane partition is a strict shifted plane partition in which the number of parts in each row is strictly less than the largest part in that row and is greater than or equal to the largest part in the next row. …

10: 36.7 Zeros

… ►Inside the cusp, that is, for

x^{2}<8|y|^{3}/27

, the zeros form pairs lying in curved rows. … ►Just outside the cusp, that is, for

x^{2}>8|y|^{3}/27

, there is a single row of zeros on each side. … ►,

y=0

), the number of rings in the

m

th row, measured from the origin and before the transition to hairpins, is given by …